Question: For real numbers $t,$ the point of intersection of the lines $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0$ is plotted.  All the plotted points lie on what kind of curve?

(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.
Answer: Solving for $x$ and $y$ in the equations $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0,$ we find
\[x = \frac{3t^2 + 3}{t^2 - 1}, \quad y = \frac{3t}{t^2 - 1}.\]Then
\[x^2 = \frac{(3t^2 + 3)^2}{(t^2 - 1)^2} = \frac{9t^4 + 18t^2 + 9}{t^4 - 2t^2 + 1},\]and
\[y^2 = \frac{9t^2}{(t^2 - 1)^2} = \frac{9t^2}{t^4 - 2t^2 + 1}.\]Thus,
\begin{align*}
x^2 - 4y^2 &= \frac{9t^2 + 18t^2 + 9}{t^4 - 2t^2 + 1} - \frac{36t^2}{t^4 - 2t^2 + 1} \\
&= \frac{9t^4 - 18t^2 + 9}{t^4 - 2t^2 + 1} \\
&= 9,
\end{align*}so
\[\frac{x^2}{9} - \frac{y^2}{\frac{9}{4}} = 1.\]Thus, all the plotted points lie on a hyperbola.  The answer is $\boxed{\text{(E)}}.$